We study the correlation functions of local operators in unitary $\textrm{ T}\bar{ \textrm{ T}}$-deformed field theories defined on a torus, using their formulation in terms of Jackiw-Teitelboim gravity. We focus on the two-point correlation function in momentum space when the undeformed theory is a conformal field theory. The large momentum behavior of the correlation function is computed and compared to that of $\textrm{ T}\bar{ \textrm{ T}}$-deformed field theories defined on a plane. For the latter, the behavior found was $\left(\frac{ \sqrt{ t}|q|}{ \pi e}\right)^{ -\frac{ tq^2}{ \pi}}$, where $q$ is the momentum and $t$ is the deformation parameter. For a torus, the same behavior is found for $|q|<<L/t$, where $L$ is the torus' length scale. However, for $|q|>>L/t$, a different behavior is found: $\left(\frac{ 2\sqrt{ t}^5q^2}{ \pi e L^3|T|^2}\right)^{ \frac{ tq^2}{ \pi}}$, where $T$ is the modular parameter of the torus. Hence, at large momentum, the correlator decays and then grows. This behavior suggests that operators carrying momentum $q$ are smeared on a distance scale $t|q|$. The difference from the plane's result illustrates the non-locality of the theory and the UV-IR mixing.
Categories: hep-th